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27 votes
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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
13 votes
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What is the origin of "two straight lines cannot enclose a space" axiom in Euclid's Elements?

It seems that the addition was made already in antiquity and motivated by the apparent gap in the proof of I.4. Rabouin in Proclus’ Conception of Geometric Space and Its Actuality writes: "...
Conifold's user avatar
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10 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
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8 votes

Has Euclid stated Cauchy's theorem?

I found it. It is Definition 10 in Book XI. https://mathcs.clarku.edu/~djoyce/elements/bookXI/defXI9.html Euclid takes the assumption of Cauchy's theorem as the definition of equality of polytopes! ...
Alexandre Eremenko's user avatar
6 votes

What happened to the original sources of Euclid's Elements?

Multiple copies of the works of Euclid's predecessors probably existed. Including similar compilations called Elements. On the opinion of people who did mathematics at the time and after Euclid, his ...
Alexandre Eremenko's user avatar
6 votes
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Euler's proof of infinite primes first since Euclid?

There is a proof by Goldbach, which appears in a letter he wrote to Euler in 1730 (a few years before Euler published his product formula for the zeta function). It is as follows: if $F_n=2^{2^n}+1$ (...
José Carlos Santos's user avatar
6 votes
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Are Euclid's theorems and proofs due to Euclid?

We do not know for sure, but most of them probably not, the proof of the Pythagorean theorem included. The current consensus on the priority of the results in Euclid's Elements is summarized by Lambek ...
Conifold's user avatar
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5 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, ...
markvs's user avatar
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4 votes
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The role of the Elements in the development of mathematics

Euclid was not the first book of this type, among his predecessors the ancient sources mention Hippocrates of Chios, Theudius of Magnesia, Leon and Hermotimus of Colofon. See this Wikipedia article ...
Alexandre Eremenko's user avatar
3 votes

How did Aristotle influence Euclid?

Heath, the famous translator of the Elements, concludes in his introduction to vol. 1 of his translation of the Elements, §"3. First Principles: Definitions, Postulates, and Axioms", that ...
Geremia's user avatar
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3 votes

How did Aristotle influence Euclid?

As Simpson clearly states in the section "Aristotelian logic" of his article, principle 3 is the law of excluded middle: $P \vee \neg P$ is true. Euclid's Elements indeed rely on classical logic ...
Mikhail Katz's user avatar
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3 votes

How did Aristotle influence Euclid?

You are right: syllogisms are not used by Euclid. More generally: “Although Aristotle emphasized the use of syllogisms as the building blocks of logical arguments, Greek mathematitians apparently ...
José Carlos Santos's user avatar
3 votes
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Who first proved necessity of Euclid's formula for pythagorean triples?

Nobody. This happens quite often, one author does something that does not measure up to modern specifications of "proving X", but, after a succession of such authors, the result is trivial ...
Conifold's user avatar
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2 votes

Euclid's use of antenaresis and Heath's commentary

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Gerald Edgar's user avatar
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