# Questions tagged [euclidean-geometry]

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

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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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### 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 ...
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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 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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### Did ancient Greek mathematicians consider numbers independently of geometry?

I am currently reading Oliver Bryne's edition of Euclid's Elements, and in The Elements many arithmetic propositions are proved geometrically, and it feels to me that numbers are always treated as ...
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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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### 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 ...
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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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### 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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### How did the notion of rigour in Euclid’s time differ from that in the 1920 revolution of Math?

I am reading about the 1900s revolution of math pioneered by figures such as Hilbert. I have seen many articles speak very fondly of these figures due to the fact they tried to study Mathematics ...
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### 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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### History of greater-than symbol used in reverse?

I was surprised to find that Oliver Byrne's 1847 marvelous The Elements of Euclid (color version)1 uses $\sqsubset$ to mean "greater than" and $\sqsupset$ to mean "less than," in ...
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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 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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### 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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### 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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### DeMorgan's commentary on Euclid's Elements

Augustus DeMorgan wrote comments on Euclid's Elements, which capture many of the most important points. Heath quotes them extensively. I cannot find any source for the original: Where can I see ...
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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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### 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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### 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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### Fibonacci and straightedge and compass constructions

In "Mathematical Thought from Ancient to Modern Times" Morris Kline claims (on page 209) that Leonardo da Pisa (Fibonacci) "showed that the roots of $x^3+2x^2+10x=20$ are not ...
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### The Original Title of "Euclid's Elements"

What did Euclid originally call his treatise of thirteen books that we now refer to as "Euclid's Elements" ? Was it "The Elements" ? Was it something else ? Does anyone know the ...
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### 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 ...
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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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### 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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### 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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A well-known proof of the Pythagorean Theorem is illustrated in the figure below: This figure shows a square with side lengths $a + b$, dissected into four right triangles (each with area $\frac 12 ... • 567 6 votes 3 answers 1k views ### What did the ratio of two magnitudes mean to ancient Greek mathematicians? My understanding is that magnitudes to ancient Greeks meant the actual line segments and plane regions (not the size of the line segment or the area of the plane region), the concept of ratio was then ... • 163 6 votes 3 answers 488 views ### How the Cross-Ratio appears in the Work of Pappus? The cross-ratio of four collinear points$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 ... • 163 6 votes 1 answer 2k 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 ... • 297 6 votes 1 answer 169 views ### When did trigonometry start using negative numbers? I'm asking this question looking at the unit circle, and thinking that greek mathematicians didn't use negative numbers. Maybe that can give enough insight into what I'm asking? 5 votes 4 answers 3k 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 ... • 5,887 5 votes 3 answers 3k 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 ... • 151 5 votes 1 answer 236 views ### Did ancient Greeks have a numerical value for the Golden Ratio Did they calculate a numerical value for the "extreme and mean ratio" or did they just have ways to construct it geometrically? If so, what value did they use and how did they calculate it? 5 votes 1 answer 190 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 ... • 163 5 votes 1 answer 154 views ### What is the origin of the "problem of Brahmagupta" of constructing inscribed quadrangle with given sides? I am looking for a source of the following construction problem: Construct an inscribed quadrangle with given sides. I know it under the name problem of Brahmagupta, but I do not know any evidence ... 5 votes 1 answer 250 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 ... • 724 4 votes 2 answers 936 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 ... • 309 4 votes 2 answers 175 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 ... • 5,887 4 votes 2 answers 165 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 ... • 141 4 votes 2 answers 506 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 votes 2 answers 231 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, ... • 1,289 4 votes 2 answers 226 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 ... • 128 4 votes 1 answer 418 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 ... • 5,887 4 votes 1 answer 1k 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 ...
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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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### 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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### 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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