It is not exaggeration if I say that Aristarchus of Samos was one of the most brilliant minds ever.
Without any doubt, he was the greatest astronomer of ancient Greece.
But, what's so extraordinary about him? He presented the first ever known heliocentric system of the Universe.
Unfortunately, his ideas about nature were completely opposite of Aristotle's works, and in that time it was almost forbidden to doubt Aristotle. Under such circumstances, Aristarchus' ideas were rejected by scientific community, and he was completely forgotten.

It would take 18 centuries until Nicolaus Copernicus presented his heliocentric model (possibly under influence of Aristarchus' previous works on that topic).

Aristarchus did some amazing work on measuring dimensions of Earth and celestial bodies, and distances  between them, without any equipment, only by using his mind...

Aristarchus of Samos, 310 BC-230 BC

On the sizes and distances of the Sun and  Moon

This is only surviving Aristarchus' work, it was part of ancient libraries and it was rewritten many times, and thanks to that it was saved. First printed version dates back in 1499 year.

In first part of this book he discussed total solar eclipse. He noticed that total solar eclipse doesn't last for so long, just couple of minutes, and concluded that because of that apparent radius of the Sun and Moon has to be the same. This conclusion can be checked simply visually.

Then, he denoted actual radius of the Sun with S, and distance Sun-Earth with D, and actual radius of the Moon with L, and distance Moon-Earth with d, then it's obvious from this image that 

                                                                     S:L=D:d

After that, he introduced the premise that during the half Moon, the Moon forms a right triangle with the Sun and Earth

Written in language of mathematics, and using the same denotation as in previous image  he got

                                                                   D:d=cos φ

where angle  φ  stands for apparent distance between the Sun and the Moon.

In this third image, positions of the Sun, the Moon and Earth, during lunar eclipse are shown. Simply put, during lunar eclipse, the Moon is in Earth's shadow. Aristarchus was observing lunar eclipse and he measured time between first contact of the Moon with Earth's shadow and the Moon completely entering Earth's shadow. Then he measured time the Moon spent being completely in Earth's shadow. He noticed that those times are equal. The conclusion is that radius of Earth's shadow, in that specific place where the Moon is entering the shadow is twice as big as the Moon's radius.

Then, he assumed that, because of huge distance between the Sun and Earth, the radius of section of the cone of Earth's shadow at the place where it is touching Earth, is the same as Earth's radius, and at the place where it is touching the Sun, same as the Sun's radius (this might sound a bit confusing, but it's clearly visible from the image above).

From this above, and from similarities of triangles abc and cde he got

                                                                            (T-2L):(S-T)=d:D

Aristarchus measured the angle we denoted with φ and he also measured apparent radius of the Sun and the Moon, let's denote this with α. Because α is really small we can say

                                                                                  α=S:D

In this moment, he had measured angles α and φ and he was able to find values of S, L, D and d (radius of the Sun, radius of the Moon, distance the Sun-Earth and distance the Moon-Earth, respectively) in respect to Earth's radius T.

In this way, Aristarchus constructed correct geometrical method to calculate dimensions and distances in Solar system expressed in units of Earth's radius. He couldn't get the exact values, but he was able to put things in real perspective, for the first time in human history.

He didn't have any instruments to measure angles and time, so his calculations were far from the truth, but his method was brilliant, and these results were of priceless significance.

His results showed that the Moon is actually big, 22 times smaller than Earth (until then people thought that the Moon is small), and that distance between the Moon and Earth is equal to 81 Earth's radii.  

The most important result was that the Sun is actually huge, at least 312 times bigger than Earth (and this result gave him another ideas), and distance Sun-Earth is equal to 1550 Earth's radii.

After this stunning results, Aristarchus was able to rise above all prejudices and to fly high to the stars, to leave Earth behind, and to see small Earth, and hundreds of times bigger Sun orbiting Earth. 

And at that moment, he got an amazing idea...

Heliocentric model

First question he had asked himself  was if Earth rotates around the Sun do we see the sky the same way?
Yes, if we assume that the Sun is static. During Earth's revolution around the Sun, the Sun is going to have apparent movement, because if Earth revolves around the Sun, we will see the Sun at different positions at the sky during the year.

Earth also revolves around its axis, and because of that we see the Sun rising and setting every day. Because of this rotation we get the illusion that the sky is moving, but actually the sky is static.

Then, he realized that change of seasons can be explained by the inclination of ecliptic relative to the celestial equator, and this can be explained with heliocentric model. It would take 2200 years for humanity to understand and embrace this idea.

His model was tested by scientific community, by asking ''If Earth really moves through space, then we should be able to notice change in positions of stars?''.
Aristarchus explained this by saying that stars are too far away, and he was right again!

The destiny of Aristarchus' ideas

Aristarchus was, obviously, in front of his time, and humanity was not ready to accept his ideas. He was forgotten, and most of his works are destroyed.
We learn how unrecognized Aristarchus was through biographical works of other scientist.

''Oh, sir, just don't bring suit against us for impiety as Cleanthes thought that the Greeks ought to lay an action for impiety against Aristarchus the Samian on the ground that he was disturbing the hearth of the universe because he sought to save (the) phenomena by assuming that the heaven is at rest while the earth is revolving along the ecliptic and at the same time is rotating about its own axis.''

Plutarch ''De facie in orbe lunae''