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The main question is why the Pythagorean theorem for right triangles:
$$ a^2+b^2=c^2$$
is such a central tool of Euclidean geometry. There are many different approaches one can take to this; I'll give it a shot.
One of the key observations is that the triangle is the most basic 'non-trivial' shape in plane (two-dimensional) geometry. Any three points - one ...
24
The truth is that we do not know. We do know of the person who is credited with the discovery, Menaechmus (c. 350 BC), a student of Eudoxus of Cnidus and a friend of Plato's, one of the most prominent mathematicians of his time. The names ellipse, parabola and hyperbola were given to them by Apollonius of Perga over a century later however. Menaechmus called ...
22
I do not agree on some details of the interpretation regarding the discovery of the irrationality of $\sqrt{2}$ as a confutation of the
Pythagoreans [...] belief that all numbers could be constructed as the ratio of 2 numbers.
My undestanding is that all "archaic" Greek mathematics shared the (implicit) assumption that, given two magnitudes, e.g. two ...
18
Yes the stories of Pythagoras that were common a few decades ago have all been been disproved, largely by Walter Burkert in Lore and Science in Ancient Pythagoreanism (1972). In short, Pythagoras never thought about any of the mathematics attributed to him. Consequently he gave no mathematical theory of music, never said all is number, and never ...
18
I'll try with some calculations : please, check it and the formulae used ...
A solid ball with a mass $m$ of $1$ kg falls (with the usual approxiamtions : no drag, etc.) with an acceleration $a$ that is about $10 \ m/sec^2$.
This means that falling from a tower $80$ meters heigh, it will touch ground after $4$ sec, with a final velocity of about $40 \ m/...
15
I doubt that many Babylonians or Greeks or any others who cared about such things ever thought Hesperus and Phosphorus were different objects any more than we think the Morning Star and Evening Star are different. Is there any evidence apart from the use of two names?
I think we today underrate what it meant to see the stars clear and bright almost every ...
14
According to this link, Legend has it that Hippasus first discovered the irrationality of $\sqrt{2}$. The second link in fact mentions a legend that held that supporters of Pythagoras murdered Hippasus -- who allegedly discovered the irrationality of $\sqrt{2}$ on a boat in the middle of the sea -- by throwing him overboard immediately after he informed them ...
12
This is a good point, I mused about it too. First, Pythagoreans and Plato had a very high minded idea of mathematics, gambling would have been seen as a lowly pursuit. This in itself does not explain it however. Plato's successors at the Academy considered mechanics lowly too, for it "uses bodies needing much vulgar manual labour, mechanics was thus ...
11
These legends do exist, and have for along time. But few if any specialist historians of the subject believe Pythagoreans discovered irrationality of $\sqrt{2}$. See:
Pythagoras vs. the idea of Pythagoras
It is very hard to judge of Greek mathematics before Euclid, let alone before Plato, as there is so little evidence. The most widely read single ...
11
The area of knowledge separates itself from philosophy as soon as a reliable method of obtaining exact knowledge in this area is invented. Thus mathematics separated from philosophy at its very beginning. In astronomy, there was an area covered by exact knowledge (based on observations) and another, speculative part.
As exact knowledge expanded, the ...
11
User plannapus points out that the proposer of the ad links to the original source, which is the first page of Diophantus’s Arithmetica, specifically the 1621 translation by Claude Gaspard Bachet de Méziriac. The right-hand side is Greek, and the left-hand side a Latin translation; the bottom seems to be the translator's commentary.
(Google Books has a ...
11
It is a strange idea that scientific laws can be only expressed with algebraic means. The Greek did discover several scientific laws. The oldest one is attributed to Pythagoras himself: it relates the length of the string to its pitch. This seems to be the oldest scientific law ever discovered. More laws were discovered in Hellenistic times: the law of ...
10
This question has been discussed several times on math overflow:
https://mathoverflow.net/questions/191909/discovery-and-study-of-conic-sections-in-ancient-greece
It also has references.
One theory is that they appeared when the Greeks started to think how to make
accurate sundial. This theory is developed in several books and articles on the subject, and ...
10
Yes.
According to this, we don't know who it was to explain the phases using a spherical model, though it was before 600 B.C:
The first person to correctly explain the phases of the Moon is lost in history. By the time Pythagoras wrote in 600 B.C., the ancient Greeks knew that the Moon is spherical and that it revolves around the Earth. The Greeks ...
10
Certainly that the Earth is spherical was a commonplace (among the educated people) at the time of Eratosthenes. Once you start traveling on sea (or climbing mountains) you immediately notice that the Earth is curved. Observations of the sky from different places confirm that and tell you that it is spherical, at least approximately (before that there were ...
10
This is a very interesting question which occupied me for a long time. I agree with L. Russo that a "scientific revolution" really happened in Hellenistic Greece. It actually happened
2 centuries before the time of Hero, at the time of the first Ptolemy's. Hero himself was probably not a great inventor, he is famous mostly because his book survived. Unlike ...
10
In short, you were taught that Aristotle was wrong because he was wrong. He didn't make a prediction, he made an observation about rock and feather, and then sloppily generalized it to all objects without a second thought. The subtle effects you are describing weren't even noticable in his time, but that a feather falls slower because it is much more ...
10
During the classical and early Hellenistic period (until 200 BC) Greeks did not use any positional system, they had their own which was decimal but not positional. The units from 1 to 9 are assigned to the first nine letters of the archaic Greek alphabet from $\alpha$ to $\theta$. Each multiple of ten from 10 to 90 was assigned its own separate letter from ...
9
The reasons are "simple".
All other axioms and postulates appeal to our "everyday experience",
at least in principle. The straight lines correspond to light rays in everyday experience.
However it was probably already recognized by Euclid, that the parallel postulate is different from the other axioms. Of course, a degree of mathematical sophistication
is ...
9
It seems that originally ratio and proportion emerged not so much "from nature", as from human activities, first practical and later more theoretical. We often find things in nature only after we already know what to look for. I will give several early examples.
In construction of the pyramids ancient Egyptian builders needed to maintain constant slope. ...
9
The Encyclopedia of Seeds: Science, Technology and Uses, edited by J. Derek Bewley, Michael Black, Peter Halmer, CABI International 2006 (Entry: History of seed research) cites some ancient descriptions along similar lines, both mythical and proto-scientific:
For example, to the Greeks, parsley was associated with death: the notorious slow germination ...
9
Euclid's Elements Book VII:
Definition 1: A unit is that by virtue of which each of the things that exist is called one.
Definition 2: A number is a multitude composed of units.
See also: Aristotle on unit (monas) and number (arithmos) and Metaphysics, Book N, 1088:
"One" evidently means a measure. And in every case it is some underlying
thing ...
8
Irrational numbers were not anathema to Pythagoreans, they never thought of them at all, or of rational numbers for that matter. The only numbers they acknowledged before, during, and after the discovery of incommensurables were positive integers. Euclid in Elements writes "ratio is a sort of relation in respect of size between two magnitudes of the same ...
8
There is an issue with "proposed" and "solved". Of all the examples the Archimedes Cattle Problem is the only one that was truly proposed and solved. The oracle of Delphi allegedly "proposed" duplication to the Athenians, but it could not have been very precise since they originally took it to mean doubling the sides. Who proposed quadrature and trisection ...
8
Yes, Aristotle was wrong about gravity. But I think it is unfair to say “that Aristotle was responsible for holding back physics for centuries”. The ones who held back physics for centuries were the late-antique and mediaeval (Christian, Muslim and Jewish) so-called philosophers who transformed Aristotelianism into an ossified dogmatic doctrine. Aristotle ...
8
(I started writing this answer from memory, but on a second look at some sources it turns out to answer the question for the statement of a rotating earth rather than that of a spherical earth... but I'll leave it here; hope the other question is still interesting to you.)
The most notable statement of a spherical rotating earth, in Indian astronomy, is by ...
8
Nothing is known for sure about the Antikythera (c.100 BC) due to its bad preservation and the lack of accompanying commentary. Perhaps the closest thing whose operation principle we do know is Heron's puppet theater (c. 50 AD) that used a complicated series of gears, rods, drums, and ropes to move small wooden figures. The contraption is powered by a lead ...
7
The fact that $\sqrt{2}$ existed and is irrational was a blow to the ancient Greeks who only believed in numbers that they could calculate to a certain degree of precision whenever required. Or in other words, they were familiar with rational numbers. The fact that others numbers existed would have carried the same sort of feelings in them as and when we ...
7
No. Aristotle was not necessarily wrong. This is in substance Carlo Rovelli's view in Aristotle’s Physics: a Physicist’s Look. As the abstracts announces it
Aristotelian physics is a correct and non-intuitive approximation of Newtonian physics in the suitable domain (motion in fluids), in the same technical sense in which Newton theory is an ...
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