Questions tagged [euclidean-geometry]
For questions about the mathematical study of shapes and space based on the works of Euclid.
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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 ...
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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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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 ...
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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?
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Golden Gnomon inside an Equilateral Triangle - Hermetic Symbol. Have you seen this object?
This has been discovered in a Jacobean publication. It houses Albrecht Durer's Vesica Piscis. There are 3 other properties that were all illustrated on the discovery, that's how I found them. They are ...
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The role of symmetry in mathematics and the half-angle formulas
It seems that the most impressive theorems of classical geometry always have to do with "half of something". Consider the following examples:
The three medians of a triangle meet at the ...
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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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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 ...
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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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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 ...
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Greater-than symbol in Byrne's *The Elements of Euclid*
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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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. ...
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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 ...
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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 ...
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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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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 ...
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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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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, ...
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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 ...
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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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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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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 ...
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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 ...
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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 ...
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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 ...
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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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