> Have you studied math?
Yes, I have a degree in it.
> And yes that is analytic geometry
No, the example you gave is not analytic geometry. Read the definition again. The explicit use of a coordinate system is essential. Algebra is not sufficient.
Yes, I have a degree in it.
> And yes that is analytic geometry
No, the example you gave is not analytic geometry. Read the definition again. The explicit use of a coordinate system is essential. Algebra is not sufficient.
> Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry without the use of coordinates *or formulae*. It relies on the axiomatic method and the tools directly related to them, that is, compass and straightedge, to draw conclusions and solve problems.
> Only after the introduction of coordinate methods was there a reason to introduce the term "synthetic geometry" to distinguish this approach to geometry from other approaches.
From wiki of analytic geometry [2]:
> In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This *contrasts* with synthetic geometry.
I know you’re using this to dismiss my case but the first line of Wikipedia hardly a standard for definition. Further down in the same article:
> *Usually* the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes three dimensions.
> As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations.
Putting it together:
1. The classic western geometry was predominantly non-synthetic, with the term “synthetic” being introduced much later in order to differentiate pure geometry from Analytic/algebraic forms
2. Use of formulae is outside the scope of pure geometry ala Euclidian style.
3. Distances and angles are geometric shapes. Trigonometry using algebra is part of analytic geometry.
4. Indian mathematicians used formulae and algebraic manipulations to arrive at geometric insights, including developing trigonometric identities and series.
5. Explicit use of coordinate system is not essential. It’s “usually” used.
6. Descartes didn’t invent the number line. He only proposed a coordinate system. Number line itself was a much later development. Descartes’ coordinates were a lot more abstract.
If using the modern coordinate system is your criteria for analytic geometry, when was it finally established?
What am I missing? Genuinely curious and happy to be completely wrong here.
[1] https://en.wikipedia.org/wiki/Synthetic_geometry [2] https://en.wikipedia.org/wiki/Analytic_geometry
> Use of formulae is outside the scope of pure geometry ala Euclidian style.
Like Heron's formula?
> Trigonometry using algebra is part of analytic geometry.
Yes, a part of analytic geometry. Distances and angles are also a part of analytic geometry.
> Explicit use of coordinate system is not essential.
Yes it is. That's literally what analytic geometry (i.e. coordinate geometry) means.
Herons formula doesn’t count because it’s the end result. Not a means to an end. You don’t need to go to Heron for that. Area of a square also has a formula and that would have made your point.
Using trigonometric identities to derive area of cyclic quadrilateral is different as algebraic equations are means and not the end.
Again, analytic literally doesn’t mean “coordinate”. It only usually does.
Also it wasn’t a wall of text to obfuscate. I sincerely laid out the full argument the best I could. Half the wall is Wikipedia quotes. The other half was itemized to make it easy to parse. But sure whatever floats your boat mate.
Yes it is. That's what analytic geometry means [0][1][2][3]. If you mean something different, use a different word.
> Herons formula doesn’t count because it’s the end result.
It came with a derivation that can be found in Heron's 60 AD book Metrica, and may have been known centuries earlier.
[0] https://www.merriam-webster.com/dictionary/analytic%20geomet...
[1] https://www.dictionary.com/browse/analytic-geometry
[2] https://www.collinsdictionary.com/dictionary/english/analyti...