Coordinate geometry was introduced by Fermat and Descartes who did not address the foundational problems that prevented the Greeks from developing this approach. This development was important for science as it allowed geometry to be quantitative and use algebraic methods, making geometry more useful for science and engineering applications through a more systematic approach than the Greek axiomatic style. While lying in bed, the mathematician Rene Descartes saw a fly on his ceiling and wondered if he could precisely describe its location, realizing vague terms like "left" were not enough and hoping to develop a precise method using coordinates.

5. History of coordinate geometry.
Co-ordinate geometry was introduced by Fermat and Descartes,
ignoring rather than solving the foundational problems which had
prevented the Greeks from taking this step (viz: the lack of any well
understood number system which could account for
incommensurable ratios).
This development is important to science because it makes
geometry quantitative and permits the use of algebraic methods.
Geometry must be quantitative for it to be useful in science and
engineering, and algebraic methods permit more rapid
development of mathematics than the less systematic (if more
rigourous) methods required by the Greek axiomatic approach to
geometry.

7. Coordinate Geometry: Get to the point!
A very famous mathematician called Rene Descartes lay in bed one
night. As he lay there, he looked up at the ceiling in his bedroom. He
noticed a fly was asleep on the ceiling. Descartes, being a mathematician
wondered if he could figure out a way of stating where exactly the fly was
on the ceiling. Obviously it has to be a precise description he thought. I
can’t really say, “To the left” or “Near the right “or “In the middle”.

10. Video of coordinate geometry.