Instances of the use and application of mathematics in Ancient Greece

Question

I am setting out to write a paper, the thesis of which only needs to have the property that it addresses "how was mathematics used in Ancient Greece".

I have begun my own investigation (i.e. excessive googling) to see what I can find, but was curious if any users of this community are already familiar with a source or sources that address interesting aspects $X$ of Ancient Greece with the property that one could argue $X$ was an application of mathematical ways of thinking.

Some first thoughts I had were that if any form of trade and barter was occurring, an argument could be presented that this required some informal notion of solving elementary algebraic equations or arithmetic. Or perhaps if any form of architecture was taking place, an argument could be presented that some informal notions of geometry / spatial relations may have been being applied.

As you can see, there are many stances one could take, and a lot of leeway in how I argue that aspect $X$ of Ancient Greece was 'using mathematics'.

Due to the fact that I do not know much about the history of Ancient Greece, I really don't know what types of things may have been occurring in their culture. Nor do I know where their notion or understanding of mathematics may have been during that time period. Of course, learning and researching both of these is part of the journey of writing this paper - so I am not expecting anyone to answer these for me, I am only hoping or inquiring as to whether anyone can point me towards some niche areas of the internet that address these topics, perhaps someone has something that may not come up in my google searching. If anyone has any sources or insight relating to the use of mathematics in Ancient Greece, that would be most appreciated!

Addendum: I did not even know enough history when I wrote the question to know how ambiguous the term 'Ancient Greece' was. The more specific time period I need to address is Classical Greece, or approximately around 500 - 200 BCE. My instructor is most likely is intending us to address instances before Euclid's formalization.

Answer 1

You should specify more precisely what you mean by "Ancient Greece". Serious applications begin in the Hellenistic period (after Alexander's conquests).

The main use of mathematics was in astronomy. Hellenistic astronomy was based on Euclid's geometry, and trigonometry was invented specially for the use in astronomy. Mathematics was also widely used in engineering: in particular in constructions of artillery, several Greek treatises on artillery survive, and in geodesy (surveying). Also in geography (I mean cartographic projections, determination of the size of the Earth, determination of geographical coordinates). Construction of sundials and water clocks also should be mentioned.

Hellenistic Greeks also had developed physics (statics, hydraulics, pneumatics, optics) which used sophisticated mathematics. This physics was used in engineering applications.

A sophisticated mechanical computer for astronomical calculations exists in a museum in Athens, probably of the beginning of the 2-nd century b.c. It is called Antikythera Mechanism (see Wikipedia).

Speaking of the earlier period (proper "Ancient Greece") the evidence of mathematics and its applications is not so abundant, but there are some remarkable pieces of indirect evidence of the use of mathematics in engineering. I mean the tunnel of Samos built in 530 bc. The tunnel exists: it was excavated in 1882. The remarkable feature is that is goes under the mountain (Castro mountain), and it was dug simultaneously from both ends! You cannot see the beginning of the tunnel from its end; the view is obscured by the mountain. Such an engineering feat requires quite sophisticated measurement which would be impossible without mathematics. The tunnel is described, for example in van der Waerden's book, Science Awakening.

EDIT. Since of all examples you seem to be interested only in this tunnel, I cite van er Waerden. It is 1 km long, and in the middle, where the diggers met, it is seen how much error did they make: it is 3 meters in vertical direction and 10 meters in horizontal direction.

Similar tunnels (dug from both ends) exist in other parts of the world, for example in Jerusalem (700 bc): but the technology is much more primitive: they just made frequent vertical wells to the surface to see where they are! And the resulting tunnel has a zig-zag shape. In the Samos case this method was impractical because of the mountain above the tunnel.