28
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 ...
- 50.3k
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 ...
- 10.5k
14
votes
Accepted
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 ...
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 (...
- 78.5k
11
votes
Accepted
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 ...
- 7,119
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 ...
- 4,070
10
votes
Accepted
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 ...
- 78.5k
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 ...
- 50.3k
10
votes
Accepted
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 ...
- 394
10
votes
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Is this a misstatement of Euclid in Halmos' Naive Set Theory book?
No error at all.
Euclid's definitions:
Definition 1. A point is that which has no part.
Definition 2. A line is breadthless length,
are probably later interpolations aimed at clarify the primitive ...
- 15.1k
9
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 ...
- 4,700
9
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 ...
- 78.5k
8
votes
Accepted
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 ...
- 78.5k
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 ...
- 78.5k
8
votes
Accepted
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 ...
- 883
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 ...
- 50.3k
7
votes
Accepted
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 ...
- 78.5k
7
votes
Accepted
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 ...
- 78.5k
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 ...
- 1,193
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 ...
- 7,779
6
votes
Accepted
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 ...
- 7,119
6
votes
When did mathematicians invent the unit circle to extend the trig functions?
The modern convention, like much of modern notation, goes back to Euler's Introductio in Analysin Infinitorum (1748), chapter VIII, there is an English translation by Blanton. The unit circle was not ...
- 78.5k
6
votes
Were there impossibility proofs for constructions in Greek geometry?
As in most such cases, nobody was "first". One can present already the Pythagorean proof of incommensurability of the side and the diagonal of a square as an "impossibility proof". ...
- 78.5k
6
votes
Accepted
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 ...
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6
votes
Did ancient Greek mathematicians consider numbers independently of geometry?
If you consider Diophantus "ancient" then the answer is "no". In his "Arithmetic" numbers are not necessarily related to geometry or physics. For Pythagoras, indeed, ...
- 892
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 ...
- 7,179
5
votes
Accepted
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 ...
- 78.5k
5
votes
Accepted
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 ...
- 78.5k
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 ...
- 78.5k
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