# 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 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 ...
1 vote
223 views

### What did Euclid mean by a straight line in his time?

The third and fourth definitions in Euclid's Elements say: The ends of a line are points. A straight line is a line which lies evenly with the points on itself. The fourth definition is usually ...
3k views

### 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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### Platonian geometry illustrated

I recently found out that a lot of Plato's work can be drawn geometrically. See the Cerritos College YouTube video "Platos Divided Line" with the description Cerritos College Professor ...
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### Were Kepler's Laws of Planetary Motion the first formal definition of an ellipse?

It seems to me that Kepler's Laws necessitate some definition of an ellipse in terms of a coordinate system. I am wondering whether Kepler's Laws mathematically defined what an ellipse is, or if he ...
1 vote
161 views

### G. Washington notes on geometry

Do you know if there is a pdf file containing President George Washington's notes on geometry and surveying somewhere in the Internet? I recall reading a few weeks ago that those notes had been ...
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### On the Euler line

Do you know of some books or papers dealing with the history of the Euler line? Was L. Euler the first mathematician that notice its existence? Are there any interesting paragraphs out there ...
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### No distance in Euclid

The mathematical concept of distance is fundamental in all mathematics and, since Bernard Riemann’s definition of manifolds, is also foundational in geometry and geometry of physics. Contrary to a ...
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1 vote
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### Were there impossibility proofs for constructions in Greek geometry?

Greek geometry was confronted with problems such as squaring the circle. Straightedge and compass constructions were unable to provide a solution, but other mechanical curves, such as the quadratix, ...
181 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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### 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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### 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 ...
126 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?
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### When did mathematicians invent the unit circle to extend the trig functions?

Is there any evidence showing that a unit circle approach was used by early mathematicians?
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### Why didn't Euclid use equations or numerals in his proofs?

I think the Elements would have been a lot more concise if he did.
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### 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.
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1 vote
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### Question about Euclid Elements book 1, definition 1

While I was reading translated into Korean version of Thomas Heath Euclid Elements, I found something weird. And I am doubting whether that translation is wrong. I will retranslate it so you guys can ...
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### 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 ...
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1 vote
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### 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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### 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 ...
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### 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 ...
151 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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### 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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### 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 ...
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### What made Euclid/Heron define line as a length without breadth and point as that which has 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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### 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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### 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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### 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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### 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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### 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 ...
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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 ...
1 vote
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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 ...
356 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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### 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 ...
261 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 ...
297 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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### 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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1 vote
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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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