reading comprehension - Synthehtic geometry


Full Member
Jul 19, 2018
Synthetic geometry is that which studies figures as such, without recourse to formulae, whereas analytic geometry consistently makes use of such formulae as can be written down after the adoption of an appropriate system of coordinates.
What is "recourse to formulae" expression meaning?
"Without using formulae" ?
Disclaimer: English isn't my first language either.
What is "recourse to formulae" expression meaning?
Look it up:

Recourse means

a: a turning to someone or something for help or protection
settled the matter without recourse to law​
b: a source of help or strength : RESORT
had no recourse left​
: the right to demand payment from the maker or endorser of a negotiable instrument (such as a check)
So "without recourse to formulae" means that you don't make use of formulas in solving a problem.

I'm not sure I entirely agree with Klein's words in Wikipedia; you can use formulas in geometry, without using coordinates.
Without using formulas means: based on a system of axioms and theorems.
"The first systematic approach for synthetic geometry is Euclid's Elements."
Examples are Thales' theorem, the intercept theorem, and similar.

Coordinates, the opposite of synthetic geometry weren't used before Descartes in the 17th century.

The Wikipedia page of "synthetic geometry" is available in 17 languages. They are usually of different content, i.e. not simple translations of each other. Maybe you can read another one, or find even your native language among them. It is always an advantage to look up the content of other languages. E.g. the German Wikipedia pages are often less theoretical and contain more proofs than the English ones.