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 Laguerre's definition of the angle via the cross ratio?

I recently read an article which said Cayley showed that affine geometry could be developed from projective geometry after he learnt of Laguerre's definition of the angle using the cross ratio. This ...
Mozibur Ullah's user avatar
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1 answer
178 views

Who was Burlet, the one from Burlet's theorem?

The Burlet's theorem is a result in Euclidean geometry, which can be formulated as follows: Theorem. Consider triangle $ABC$ with $\angle{C}=\gamma$. Let $P$ be the point where the incircle touches ...
Emmanuel José García's user avatar
8 votes
1 answer
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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 ...
SRobertJames's user avatar
3 votes
0 answers
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History of right hand rule

I am curious to know when the right-hand-rule for vector product was established and used consistently in mathematics. I read here Who gave right hand thumb rule for circular loop of current ...
Sofia Tirabassi's user avatar
6 votes
2 answers
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First appearance of the "four triangles and a square" proof of the Pythagorean Theorem

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 ...
mweiss's user avatar
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7 votes
1 answer
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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 ...
DDS's user avatar
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Was the small Desargues Theorem known to ancient Greeks?

My question concerns the classical Desargues Theorem and its simplest version The small Desargues Theorem: Let $A,B,C$ be three distinct parallel lines and $a,a'\in A$, $b,b'\in B$, $c,c'\in C$, be ...
Taras Banakh's user avatar
5 votes
1 answer
234 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?
Adrien Hingert's user avatar
10 votes
3 answers
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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 ...
tryst with freedom's user avatar
1 vote
0 answers
146 views

How did Euclid arrive at the law of reflection $r=i$?

The law of reflection, $r=i$, is attributed to Euclid. In his Optics text he refers to it, at the end of Page 360 in relation to Graph 19, and says he has demonstrated it in his Catoptrics. But I do ...
Maesumi's user avatar
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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 ...
Frunobulax's user avatar
2 votes
1 answer
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Searching for book about non-Euclidean geometry that recapitulates the First Book of the Elements

I am looking for a specific book on non-Euclidean geometry that I read in my undergraduate years. The unique characteristic of this book is that the first part of the book started by re-proving in ...
ltcomdata's user avatar
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1 answer
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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 ...
Joseph O'Rourke's user avatar
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2 answers
486 views

Newton's Corollary #1 to the Laws of Motion (Principia)

I'm currently working through selected portions of Newton's Principia, but I'm already stuck in trying to understand his explanation for the first corollary (i.e., Corollary I) to the laws of motion. ...
Andrew R.'s user avatar
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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 ...
Anton Petrunin's user avatar
2 votes
2 answers
371 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 ...
Euclid Looked On Beauty Bare's user avatar
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2 answers
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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 ...
Thomas.M's user avatar
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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 ...
user1801060's user avatar
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1 answer
131 views

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 ...
Clark Merala's user avatar
1 vote
1 answer
182 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 ...
Jamai-Con's user avatar
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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 ...
Jamai-Con's user avatar
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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 ...
massimo's user avatar
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1 vote
1 answer
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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, ...
elias1952's user avatar
4 votes
2 answers
228 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, ...
Prime Mover's user avatar
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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 ...
Chrystomath's user avatar
6 votes
1 answer
211 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 ...
sqrt_eric's user avatar
6 votes
1 answer
166 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?
A Citizen of The World's user avatar
3 votes
1 answer
1k views

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?
Dom Turner's user avatar
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0 answers
122 views

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.
Siddhartha's user avatar
3 votes
1 answer
2k 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.
Siddhartha's user avatar
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 ...
Vito's user avatar
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4 votes
2 answers
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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 ...
Chaim's user avatar
  • 309
1 vote
1 answer
202 views

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. ...
Vito's user avatar
  • 37
4 votes
2 answers
173 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 ...
José Carlos Santos's user avatar
2 votes
1 answer
441 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 ...
mathmathmath's user avatar
5 votes
1 answer
189 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 ...
bblohowiak's user avatar
4 votes
2 answers
164 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 ...
Tomas By's user avatar
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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 ...
Widawensen's user avatar
2 votes
2 answers
921 views

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 ...
Sensebe's user avatar
  • 675
10 votes
1 answer
296 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 ...
harpersferry's user avatar
0 votes
1 answer
194 views

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'...
j4dk's user avatar
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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 ...
Marcin Ciura's user avatar
0 votes
2 answers
179 views

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 ...
Keshav Srinivasan's user avatar
3 votes
1 answer
170 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 ...
José Carlos Santos's user avatar
4 votes
1 answer
81 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 ...
Patrick's user avatar
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3 votes
1 answer
248 views

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 ...
Calum Gilhooley's user avatar
1 vote
0 answers
169 views

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 ...
Hans-Peter Stricker's user avatar
4 votes
2 answers
500 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: ...
Hans-Peter Stricker's user avatar
0 votes
1 answer
117 views

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 ...
Hans-Peter Stricker's user avatar
4 votes
1 answer
517 views

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 ...
Hans-Peter Stricker's user avatar