Questions tagged [geometry]
Branch of Mathematics about the properties of the shapes, their similarities and transformations in the space.
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First occurrence of hyperboloid paraboloid
The ancient greeks considered surfaces such as cones, but did they study the hyperbolic paraboloid? What is the first occurrence of such surface in history?
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The role of the Elements in the development of mathematics
The Elements are often regarded as the cornerstone of the axiomatic approach to mathematics. However, mathematical textbooks have served as the foundational pillars upon which writing style, language, ...
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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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Kepler's Mysterium Cosmographicum with regular polygons: in which nesting order?
Kepler tried to use regular polygons before using the 3D platonic solids in his Mysterium Cosmographicum.
My question is: which regular polygons did Kepler try to use and in what nesting order?
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Origin of $V_a$ (median) notation
My question about median of a triangle.
The English equivalent of the Turkish word "kenarortay" is "median". In English-language geometry sources (like books or web pages), the ...
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A text or YouTube channel with a comparison between pre-Cartesian with post-Cartesian mathematics
Is there some good book or YouTube channel that make a good comparison/distinctions between the mathematics before René Descartes, with the mathematics after Descartes?
In the short article "The ...
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Question on Gauss's geometric interpretation of "spherical functions"
In the physics chapter of his biography of Gauss, W.K. Buhler writes the following:
Expansions into series are frequent and important in potential theory. So it does not come as a surprise that ...
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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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Leibniz conjecture that geometry is a form of algebra
Hermann Grassmann (1840s) verified the Leibniz conjecture that geometry 1s a form of algebra, showing that the geometric figures themselves are algebraic entities, because they are subject to definite ...
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When and why Cantor-Hume principle was universally adopted in set theory instead of Euclid's principle?
In this answer and the comments Joel David Hamkins talks about a conflict between Cantor-Hume principle and Euclid's principle. He writes:
This principle [Cantor-Hume] is often defended as a ...
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Did Riemann invent the Riemann curvature tensor?
I'm pretty sure they weren't using tensors in the modern sense at that point, but to what extent did Riemann lay out the structure or significance underlying his eponymous tensor? In his habilitation ...
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Why does Eratosthenes method for calculating the circumference of the Earth requires the city of Alexandria and Syene to be in the same meridian?
I'm reading a book where the author claims that in order for the method of angles and proportions used by Eratosthenes to work, the two cities would have to be located in the same meridian, or at ...
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Why is bachelors' unknotting called as such and who discovered it?
Bachelors' unknotting is a way to show that all tame knots are isotopic to the unknot, by tightening a knot to a point. Why is it called 'bachelors' unknotting'?
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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 was Gauss's algorithmic method to solve a certain "nearest neighbour search" problem in multi-dimensional euclidean space?
In his 1829 paper on a new formulation of mechanics, Gauss presented his principle of least constraint, which parallels previous formulations of analytical mechanics and provides a new point of view ...
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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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How to build a protractor without a protractor?
We all know how to use a protractor, it is taught in elementary school. However, I was wondering what type of knowledge is required to build one from scratch.
For instance, was the understanding of $\...
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Is there any book dealing with the history of both astronomy and geometry?
I am preparing an introductory course on the history of astronomy and geometry for intermediate students. The intention is to teach Copernican heliocentrism, Kepler's laws of planetary motion & ...
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What was the pre-Renaissance proto-perspective scaling technique for painting square tiled floors?
I read once (I don't remember where exactly) about an early technique (Early Renaissance) to draw a square tiling floor in perspective. The next row in the drawing is done multiplying the previous one ...
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When did we first know the $11$ ways of opening up a cube?
Take a box in the shape of a cube and cut $7$ of its edges so that all the faces stay connected and flatten it out onto a table. It turns out, there are $11$ distinct meshes that emerge: https://math....
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Origin of the term "affixe"/"affix" in the geometric treatment of complex numbers
In current French mathematical tradition, when introducing complex numbers, it is common to hear about "complex plane of Argand-Cauchy".
What is particular in French treatment, it is the ...
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Why is the letter $b$ used to represent the y-intercept in the equation of straight line?
The slope-intercept form of a non-vertical line is $y=mx+b$. I have been told that the slope is called $m$ because it is the first letter of the French word for mountain. But why is there the letter $...
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How to give a meaningful interpretation to Gauss's notion of "oriented area" of self-intersecting geodesic polygons?
Gauss's notion of "oriented area" of figures characterizes the notion of two-dimensional content in a way that enables self-intersecting geodesic polygons (the term "geodesic" ...
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Discussions of why the Greeks "squared the..." geometrically
The Wikipedia article on squaring the circle has a relatively good history of their efforts on this particular problem, but it fails to mention why the Greeks were interested in this methodology on a ...
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Research on Pre-Columbian Polynesian geometry?
Has there been any historical/cultural anthropological research into how Polynesian cultures understood geometry before contact with Europeans? In part what I am interested in is how did the dealing ...
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Why is this notation used to define points in (elementary) analytic geometry?
I have always found strange that in elementary analytic geometry points are defined by their names followed by their coordinates, for example:
"Find the distance between $A(5, -3)$ and $B(2, 1)$....
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Explanation request for the terminology and notation employed by Gauss in his major 1843/6 treatise on Geodesy
Background:
In his 1827 treatise on differential geometry, Gauss in his "theorema egregium" proved that the curvature of a surface is an intrinsic invariant; it doesn't change under ...
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How was longitude determined in the 1700s?
I'm going through the journals of Alexander Mackenzie (ca 1790) and I came across this passage:
I gather that he's determining his latitude and longitude but I'm not clear on what units he's using, ...
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Did the ancients have the concept of dimension?
Plainly, they knew what a circle and sphere were and also a square and cube; but did they discuss the idea that a sphere was analogous to a circle but in 3 dimensions or similarly the analogy between ...
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Why did Columbus think the Earth was much smaller than it is?
Columbus set off on his westward journey to Asia, believing the Earth was much smaller than it is.
There was some apparent disagreement about the size, and Columbus was staking his life on it. Why ...
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Why 360° is assigned to circle full turn ? Not any other number? [duplicate]
Please look at this question
https://math.stackexchange.com/posts/comments/9011243?noredirect=1
A user comment this so I thought of asking here
You mean why did we decide on using 360 degrees? I don'...
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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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Were units of area/volume always in terms of squares/cubes?
Throughout our known history of geometry were the units representing areas and volumes always in terms of squares and cubes respectively? Take ancient Egyptian formulas as an example, the fact that ...
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Historicity of Euclid. Looking for the references of Euclid in ancient texts that has survived [duplicate]
The general consensus is that Euclid was a real historical figure. Wikipedia https://en.wikipedia.org/wiki/Euclid concludes on the hypothesis that Euclid was not a real person,
"This hypothesis ...
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What is the History of Coordinates?
Can someone give a history of coordinates? When did coordinates first appear, which was the first ever coordinate system? In which field were they used?
user14943
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Analysis of Moscow Mathematical Papyrus problem no. 10
This problem is giving me headaches for quite some time now I guess mainly because I am not a mathematician, but I would very much like to know if there is anyone here that has any deeper insight into ...
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Help understanding Egyptian circle
I was reading this Wikipedia page searching for the Egyptian area of circle and there is a following picture there:
Trying to understand what is meant by this since it is under the "Area" ...
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Explanation of the main points in Gauss's resultant calculus
After he read Mobius's 1827 treatise on the "barycentric calculus" (according to Gauss's own testimony, he read this treatise only in 1843), Gauss wrote down several unpublished notes on ...
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Which was the first coordinate system? And why was it used?
I've been researching the motivation behind the invention of the Cartesian Co-ordinate system. But I was wondering, which coordinate system was the first to be invented? Was it the Spherical Co-...
user14943
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What does "organic" mean in old texts when describing plane curves and their construction?
I've been reading about 17th and 18th century geometers and their research into plane curves, especially algebraic curves. A term that comes up frequently is "organic". By context it seems ...
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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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How did Roger Cotes come up with logarithm form of Euler formula?
I have been trying to get my head around how Roger Cotes first discovered Euler Formula.
I knew how Euler did it, but I wanted a new perspective, especially from someone who discovered it earlier.
...
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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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Beltrami's Essay on the Interpretation of non-Euclidean Geometry
I am reading the Essay of the title written by Beltrami in Italian and I found a specific point of the essay which in my opinion could be fully clarified only if compared with its translations. At the ...
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Origin of "Inverse Pythagorean Theorem"
There is a lot of information on the history of the Pythagorean theorem, but not much on its closely related cousin; the Inverse Pythagorean Theorem. Would appreciate any resources on the the history ...
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Does Hadamard claim that Pascal could have discovered non-Euclidean geometry?
In The psychology of invention in the mathematical field, p. 53, Hadamard makes the following claim:
Is his point that there must, in any axiomatic theory, be undefined terms, and if you write the ...
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Fourth powers and quartic equations before Descartes
How did mathematicians interpret quartic equations and fourth powers before Descartes propose to perform elementary arithmetic on line segments?
I ask this because it seems strange to me that ...
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Where can I find the complete papers of abstracts published by P. G. Tait in Proc. Roy. Soc. Edinburgh in 1880?
I am interested in looking up P. G. Tait's flawed proof of the four-colour theorem, published in 1880. The citation that I have seen is:
P. G. Tait, On the colouring of maps, Proc. Roy. Soc. ...
user13915