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61

This device was invented by a Jewish Rabbi, Levi Ben Gershon. It was used to measure the angular distance between two stars or, in general, any pair of celestial bodies. Ptolemy lived 1000 years before the invention, so this is only the product of the artist’s fantasy.


56

It is called "Jacob's staff". It was an old astronomical tool used for trigonometric purposes.


30

One early invention for storing energy was a basin above the level of the river. It was filled with water when the water in the river was high, and then, when the water in the river was low, it was allowed to flow to the fields from the basin. Such a basin could also be filled manually. Such devices were used in ancient Egypt, as a part of their irrigation ...


20

This is probably not what you were thinking of but "the earliest invention that allowed energy to be stored and released after a delay even it's just a short time" was a stone. I can store energy in it briefly by flinging it at your head (kinetic). I can store energy in it even longer by putting it on a cliff above your head (potential). Stone ...


15

And this tool has been known under many other latin names than baculus Jacob (or Jacob's staff): radius astronomicus (astronomic ray), crux geometrica (geometrical cross), revelatorem secretorum (secret decipherer). This one is the hand-held version, a mounted version exists. A collection of studies on Gersonides (or Levi ben Gershon) is found in G. ...


9

Agriculture The purpose of farming is to harness the freely available energy from the sun, and convert it into a form that can fuel the human body. Agriculture enabled significant food surpluses, which is simply a form of energy storage. Similar to how a flywheel stores mechanical energy which can be used later, food surpluses store chemical energy which can ...


5

Rigorous notion of limit for special cases arose in the work of Eudoxus and Archimedes, when determining the length of a circle, volume of the pyramid etc. (The work of Eudoxus did not survive, we know about it from Euclid and ancient historians of mathematics). The argument to find these limits was called the "method of exhaustion". It is ...


5

No, they did not. Several methods were proposed but they do not give "correct" distances. Of course, all depends on the exact meaning of the word "measure" and "correct". But their estimates were orders of magnitude away from the true numbers. Some details are given here: Historical knowledge of Distance of Earth from Sun and also here the related ...


5

Ptolemy knew about the Moon's parallax (he explains it in section 11, Ch. V of Mathematical Syntaxis). To measure it he invented the "parallactic instrument" described in section 12. Section 13 is dedicated to determination of Moon's distance, where he explains his observation in great detail. Roughly speaking he computes the Moon's geocentric position in ...


4

From "Queries and Answers," Isis 41, no. 2, linked to by sand1 in the comments above, we see that the proof of the sphericity of the earth based on the fact that the masts of a distant ship are visible above the horizon when the body of the ship is no longer visible ...has been attributed to Aristotle in De Caelo, but it does not appear there. It does, ...


4

I dug up some text from a translation of "on the heavens." There are similar disputes about the shape of the earth. Some think it is spherical, others that it is flat and drum-shaped. For evidence they bring the fact that, as the sun rises and sets, the part concealed by the earth shows a straight and not a curved edge, whereas if the earth were ...


4

Historians believe that the extant version of Problemata was not penned by Aristotle personally, but “while the Problemata is not the genuine Aristotelian work, it nevertheless contains an element derived from such a work”. Problemata XXXII.5 discusses breathing under water, including the oft quoted passage interpreted as referring to a diving bell:”Why do ...


4

According to this page, charcoal was in use circa 3750 BCE. That's an energy storage medium, although perhaps not the class of energy-release you were thinking of.


3

How about good-old fashioned pit traps? If 'stored energy' is something where you input work ahead of time and then it is expended at a later date for a use, a pit trap is a perfect example. While digging the pit, you are removing a great deal of material, and effectively 'storing' gravitational potential energy in the difference between the height of the ...


3

As @Andrei Kopylov noticed, epicycle theory is not the theory of Fourier series of a periodic function. Still this is called (generalized) Fourier analysis. Such functions are called almost periodic or quasi-periodic, and they expand into generalized Fourier series of the form $$\sum c_k e^{i\lambda_k t},$$ with arbitrary real $\lambda_k$, which is also ...


3

For Aristotle : Problemata Book XXXII, 960b,30-31 : “divers to respire equally well by letting down a cauldron”. Parts of Animals, Book II, 659a,5 : “Some divers, when they go down into the sea, provide themselves with a breathing-machine, by means of which they can inhale the air from above the surface while they remain for a long time in the water.” For ...


2

The sentiments you are describing are most likely those attributed to Plato. Although not primarily a mathematician, Plato is the figure who is credited with drawing the distinction between pure and applied mathematics - drawing a line between the theoretical and the computational aspects of mathematics, and instilling this in his followers. According to ...


2

Much of the ancient transmission went through Greece. Astronomical (and other) knowledge was transmitted from Egypt to Greece at least by the time of early Pythagoreans (c. 500 BC), likely earlier. According to some sources, Pythagoras and Democritus visited Egypt personally to study with the priests. There might have been some early transmission from ...


2

Tree limb. Before humans became ground bound, they used tree limbs to catapult themselves to the next tree. But seriously, probably a stone, then a stone attached to a stick, then a stone attached to a stick shot from a string attached to a stick. Do I get voted off the island? I just got here.


1

I agree to candied_orange that it might be a stone, but for different reasons. There still is a practice among nomadic people to heat stones in a fire [aka store energy] and then transfer them into a container (e.g. the fur of an animal filled with water) to cook/boil the contents [release the energy]. Here are two sources I could find: Travel report from ...


1

This is very popular myth, but it is not true: Ptolemy's epicycles are not Fourier analysis! Fourier series can indeed approximate an arbitrary periodic function. And you can approximate an arbitrary motion of period $T$ by the series of epicycles. The first is just a circular motion of period $T$, the second is an epicycle with period $T/2$ and so on. But ...


1

Here are some Roman Denari. Moon and stars. Looks like points on the stars.


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