3
votes
1answer
42 views

Who first proved that every triangle has a circumscribed circle?

Wikipedia only mentions that it follows from the Cartesian equation for a circle: $\left(x - a \right)^2 + \left( y - b \right)^2=r^2$ https://en.wikipedia.org/wiki/Circumscribed_circle#cite_note-1 ...
41
votes
7answers
5k views

What is the oldest open problem in geometry?

Geometry is one of the oldest branches of mathematics, and many famous problems have been proposed and solved in its long history. What I would like to know is: What is the oldest open problem in ...
7
votes
2answers
132 views

History of the power of a point with respect to a circle

There is a concept of a "power of a point with respect to a circle". If one has a point which is distance $d$ away from the centre of some circle and that circle has radius $r$ then the power of this ...
6
votes
1answer
207 views

Where does the “Visual Multiplication” technique originate from?

There is a geometric technique to perform multiplication of numbers. But as the internet goes, it is hard to figure out who deserves the credit. What I've heard is A mayan technique From Vedic ...
0
votes
0answers
19 views

Homogeneous Spaces: The Erlangen Programme

This is a wholly a question of mathematics history. The Klein Erlangen programme is most pithily, if a little tersely, described in modern wording as a homogeneous space: a topological group acting ...
4
votes
2answers
164 views

Why did the ancients hate the Parallel Postulate?

I am reading this book, Gödel's Proof, by James R. Newman, at location 117 (Kindle), it says, For various reasons, this axiom, (through a point outside a given line only one parallel to the line ...
3
votes
1answer
51 views

Why half coversed or coversed trigonometric functions are being deprecated?

As you can see here there are some names for some trigonometric functions that I can't find in any text or math related papers today. In my opinion this kind of approach will also make it easier to ...
2
votes
2answers
157 views

Which small area of mathematics had fully developed already and thus no more research in this area?

Which small area of mathematics had fully developed already and thus no more research in this area? For example, no more PHD research in Euclidean Geometry anymore.
3
votes
0answers
73 views

History of incenter and Euler line

It is easy to see that if a triangle is isosceles, then its incenter lies on its Euler line. Who first proved the converse of this result and what technique was used? (See the post "The incenter and ...
5
votes
1answer
164 views

Could Euclid have bisected a line segment without his method of superposition?

In Book I Proposition 10 of the Elements, Euclid performs the bisection (i.e. finding a midpoint) of a line segment. In the course of doing so, he uses Book I Proposition 4, the Side-Angle-Side ...
5
votes
2answers
262 views

Proofs without words of some well-known historical values of $\pi$?

Two of the earliest known documented approximations of the value of $\pi$ are $\pi_B=\frac{25}{8}=3.125$ and $\pi_E=\left(\frac{16}{9}\right)^2$, from Babylonian and Egyptian sources respectively. ...
2
votes
2answers
48 views

translation of Pasch “Vorlesungen über die neuere Geometrie”

Has Pasch "Vorlesungen über die neuere Geometrie" ever been translated?
3
votes
1answer
290 views

Is Euclid's Fourth Postulate Redundant?

Euclid's Elements start with five Postulates, including the fifth one, the famous Parallel Postulate. Less well, known however is the Postulate that forms the basis for motivation behind the fifth: ...
2
votes
1answer
133 views

Historical meaning of determinant

Some time ago, I published the following question Geometric meaning of the determinant of a matrix on the geometric meaning of determinant. Usually, on the books of algebra, the determinant is ...
0
votes
1answer
215 views

Beginnings of Greek Mathematics

For another proof of the pythagorean theorem, consider right triangle ABC (with right angle at C) whose legs have length a and b and whose hypotenuse has length c. On the extension of side BC pick a ...
15
votes
1answer
294 views

Motivation for the study of amoebas.

What was the primary motivation for the study of the amoebas?
7
votes
4answers
230 views

Is there an established notation, either modern or historical, for any unit of measure which is then further subdivided into 360 degrees or parts?

This question about notation is simple as dirt, but would be useful for me regardless, because of some work that I'm doing in music theory. Basically, while there's a notation for subdividing the ...
2
votes
3answers
108 views

How prove this inequality for $a,b,c,d$ are real numbers

let $a,b,c,d$ are real numbers,show that $$2\sqrt{a^2+c^2}+\sqrt{a^2+c^2+3(b^2+d^2)-2\sqrt{3}(ab+cd)}+\sqrt{a^2+c^2+3(b^2+d^2)+2\sqrt{3}(ab+cd)}\ge6\sqrt{|ad-bc|}$$ This problem is creat by China's ...
1
vote
2answers
355 views

What is the initial reason to define the evolute of a curve?

The evolute of a curve is defined as the envelope of the normals or as the locus of the center of the osculating circle. What is exactly "the envelope of the normals" ? What is the reason to ...
2
votes
1answer
146 views

How do we know $\pi$ is a constant? [duplicate]

How did the ancient Greeks discover that the ratio of a circle's circumference to its diameter is constant? It does not seem so intuitive. Thanks!
6
votes
3answers
291 views

History of Conic Sections

Recently, I came to know that ancient Greeks had already studied conic sections. I find myself wondering if they knew about things like directrix or eccentricity. (I mean familiar with these concepts ...
1
vote
0answers
86 views

How was the isoperimetric inequality formulated?

I'm tyring to understand how the isoperimetric inequality came into existence. It seems like finding the region which yields maximum area when enclosed by a curve of fixed length is an old problem. ...
4
votes
3answers
2k views

Why do we have 360 degrees in a circle and why we need radians? [duplicate]

I have two related questions: 1- Why do we have 360 degrees in a circle? 2- I have seen in most of the mathematical concepts, angle is expressed in radians not in degrees. Why was radian ...
2
votes
0answers
283 views

On the geometric arguments used in Newton's *Principia Mathematica Naturalis Philosophae*

When one reads Newton's Principia Mathematica, one is immediately aware of the complexity of the synthetic geometry that he uses to prove his propositions. This I understand because all of the ...
9
votes
1answer
747 views

ancient concepts and modern concepts

Is there an extant published expository account, comprehensible to all mathematicians, of the conceptual differences between ancient Greek mathematical concepts and modern ones? I have in mind things ...
40
votes
5answers
3k views

Why is a full turn of the circle 360°? Why not any other number?

I was just wondering why we have 90° degrees for a perpendicular angle. Why not 100° or any other number? What is the significance of 90° for the perpendicular or 360° for a circle? I didn't ever ...
5
votes
2answers
334 views

the definition of the area of a surface

When we say the area of a rectangle is the product of the length by the width is it a definition based on geometric intuition or is it a result? I know it is a result that we can find after defining ...
8
votes
3answers
753 views

Volumes of cones, spheres, and cylinders

Given a sphere with radius r, a cone with radius r and height 2r, and a cylinder with radius r and height 2r, the sum of the volume of the cone and sphere is equal to the volume of the cylinder. If we ...
8
votes
1answer
339 views

who first defined a tangent to a circle as a line meeting it only once?

From googling, it seems commonly believed that Euclid did this, but it seems nowhere in Euclid does he even state this property of a tangent line explicitly. Rather Euclid gives 4 other equivalent ...