27 votes
Accepted

Did ancient Greek mathematicians consider numbers independently of geometry?

The answer is yes. There was a split. First of all, for the Greek mathematics (and very long after them) "numbers" were integers. "Rational numbers" were called fractions, and no ...
Alexandre Eremenko's user avatar
14 votes

How did the notion of rigour in Euclid’s time differ from that in the 1920 revolution of Math?

For example, the very first proposition: Construct an equilateral triangle $ABC$, where one side $AB$ is given. Euclid says Draw a circle with center $A$ and radius $AB$. [By Postulate 1] Draw the ...
Gerald Edgar's user avatar
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13 votes
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DeMorgan's commentary on Euclid's Elements

De Morgan's "Short Supplementary Remarks on the first Six Books of Euclid's Elements" is contained in the Companion to the (British) Almanac for the year 1849, pp.5–20, published by the ...
Alexander Campbell's user avatar
12 votes
Accepted

What theorem of Sophus Lie on the number of geometries is H. Poincaré referring to?

Poincare refers to the Lie's solution of the so-called problem of space, a.k.a. the Helmholtz , or Riemann-Helmholtz, or Helmholtz-Lie problem of space, which amounts to characterizing all manifolds (...
Conifold's user avatar
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11 votes
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Why didn't Euclid's Elements treat conic sections?

While Elements contains no reference to conic sections it does define angled cones, given as definition 18 of Book XI, and it examines some of their properties in Book XII. History identifies ...
nwr's user avatar
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11 votes
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Fibonacci and straightedge and compass constructions

In the Flos (Flos Leonardi Bigolli Pisani super solutionibus quarundam questionibus ad numerum et ad geometriam, vel ad utrumque pertinentium), Fibonacci reinterprets in algebraic form the geometric ...
user6530's user avatar
  • 3,850
10 votes

How did the notion of rigour in Euclid’s time differ from that in the 1920 revolution of Math?

The main difference is that Mathematical logic and set theory did not exist at the time of Euclid. (The Logic of Aristoteles is still very far from mathematical logic created in 19th century). As a ...
Alexandre Eremenko's user avatar
10 votes
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The Original Title of "Euclid's Elements"

It is "Στοιχεῖα" [ Stoikheîa ] in Ancient Greek. https://en.wikipedia.org/wiki/Euclid%27s_Elements : Euclid's Elements (Ancient Greek: Στοιχεῖα Stoikheîa) is a mathematical treatise ...
Prem's user avatar
  • 396
9 votes
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History of greater-than symbol used in reverse?

Byrne's symbols are variations of Oughtred's, contamporaneous with Harriot's, see Cajori, History Of Mathematical Notations, vol. I, p.192. They were adopted by Barrow, Newton's teacher, in his ...
Conifold's user avatar
  • 75.2k
8 votes

Why is calculus missing from Newton's Principia?

Although this question and the answers now have some age to them, I suggest that it's important not to overlook the mythical character of the assumption that underlies this question. The question ...
terry-s's user avatar
  • 4,375
8 votes

What did the ratio of two magnitudes mean to ancient Greek mathematicians?

Fowler's Ratio in Early Greek Mathematics is a standard reference on the subject, see also his book Mathematics Of Plato's Academy (both are freely available). Book V, Definition 3 of Euclid's ...
Conifold's user avatar
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8 votes
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What is the history of angle quintisection (division into five equal parts)?

Not much history to it, I am afraid. It seems that methods of trisection rather obviously (to those who considered them) applied to quintisection as well, so the problem was of little theoretical ...
Conifold's user avatar
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8 votes
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What caused or contributed to Euclid's Elements and Synthetic Geometry falling into disfavor?

The contest between synthetic and analytic methods in geometry predates Hilbert and even calculus, one can trace its origins to Vieta's algebraic conversions of geometric problems that streamlined ...
Conifold's user avatar
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8 votes
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Did ancient Greeks have a numerical value for the Golden Ratio

First of all, Greeks were not fascinated with Golden ratio as we are. Modern golden-ratio hype started about from the time of Leonardo da Vinci. Second, Greek mathematicians were not very interested ...
Alexei Kopylov's user avatar
7 votes

Compass and straightedge: why?

The real reason is probably that straightedge and especially compass are the simplest, most primitive instruments, and also easy to make. At the same time they are quite accurate. (Straightedge is not ...
Alexandre Eremenko's user avatar
7 votes
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How did the use of the word "origin" become commonplace in geometry?

A good place to look for such things is Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics, where we read: "Boyer (page 404) seems to attribute the term origin to Philippe de ...
Conifold's user avatar
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7 votes
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What were the applications of conic sections before Kepler?

There is no doubt that ancient Greeks were primarily interested in conic sections for their intra-mathematical uses. Even aside from the Delic problem, the legendary motivation of the discoverer ...
Conifold's user avatar
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6 votes
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Who classified plane isometries first?

Coolidge (1940, p. 273) gives a proof which ends: (...) Hence any real transformation of the Euclidean plane which keeps distances invariant is either a rotation, a translation, or the product ...
Francois Ziegler's user avatar
6 votes

When and who was the first mathematicians to prove rigorously that $\sqrt[3]{2}$ was impossible number?

Gauss considered the algebraic background behind straightedge and compass constructions, and from his work it is clear that e.g. $\sqrt[3]{2}$ is impossible to construct, as it can't be expressed via ...
vonbrand's user avatar
  • 555
6 votes

Did Dieudonné say "Euclid must go!" or "Down with Euclid! Death to triangles!"?

I hope a version in French may shed some light on this question. In Le Séminaire de Royaumont 1959-1979, here is apparently a verbatim of the words of Jean Dieudonné. The previous link ...
Laurent Duval's user avatar
6 votes
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When Indian mathematicians learn of Euclid's Elements?

According to The Hindu Business Line, quoting the scholar TA Sarasvati Amma: It was only in the 18th century, nearly 2,000 years after active contact of Indians with the Greeks, that Euclid’s ...
nwr's user avatar
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6 votes
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Were Kepler's Laws of Planetary Motion the first formal definition of an ellipse?

No. Conic sections were studied in Greece since the IVth century BC. And Apollonius of Perga (c. 240 BC – c. 190 BC), who coined the term ellipse, wrote a treatise on conic sections (Conics) in eight ...
José Carlos Santos's user avatar
6 votes
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Who was Burlet, the one from Burlet's theorem?

I am unable to find any references to a "theorem of Burlet" in a print publication regardless of language, e.g. Satz von Burlet, théorème de Burlet. All online resources referencing this ...
njuffa's user avatar
  • 6,496
5 votes
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History of the quadrature of curvelinear figures prior to the middle ages

Pappus in Mathematical Collection, book IV "squared" the Archimedean spiral, the spiral is to the circle enclosing it as 1:3. The curve is obtained by composing uniform linear and circular motions in ...
Conifold's user avatar
  • 75.2k
5 votes

Who discovered integer triangles with one angle trisecting another?

This is a neat observation. I was unable to find historical references for it in particular, but there is a rich history of solving this type of problems. The problem is clearly reminiscent of finding ...
Conifold's user avatar
  • 75.2k
5 votes

On Einstein's proof of the so-called Pythagorean theorem

Regarding II (c): Strogatz/Schroeder does offer a string of connections to Einstein through Shneior Lifson and Ernst Straus, one of Einstein's assistants at IAS. But it would be hard to connect ...
Brian Hopkins's user avatar
5 votes
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What was the relation between Euclid's points and Democritus' atoms?

They were the opposite of close. Geometers and atomists were bitter ideological enemies since before Euclid. The reason for such high passions was that Greek view of geometry was different from the ...
Conifold's user avatar
  • 75.2k
5 votes

What made Euclid/Heron define line as a length without breadth and point as that which has no part?

As mentioned in the comments, the definitions were probably inserted by later authors, like Heron, for didactic purposes and/or for compliance with Aristotelian ideas about proper scientific ...
Conifold's user avatar
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5 votes
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Did Ostrogradsky dismiss Lobachevsky's book on non-Euclidean geometry "because the world is obviously Euclidean"?

Ostrogradsky is well-known for his negative reaction to Lobachevsky's work, but in his signed reviews, at least, his complaints were different. Lobachevsky, in contrast to Ostrogradsky, was not a good ...
Conifold's user avatar
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