Comment by jimsimmons | Hacker News Reader

Comment by jimsimmons | Hacker News Reader

Parent Full thread

Hard to give a concrete source because it’s simply not there. See my comment in the sibling thread. I can make a case and you can judge for yourselves.

The key insight in analytic geometry is that geometric shapes and algebra are facets of the same underlying concept. This allowed Descartes to develop coordinate system. This in turn allowed development of calculus and everything else. This was a striking discovery in the western world.

The practice of using equations for geometry however was very common in India. Brahmagupta was the first to have a formula for area of cyclic quadrilateral in 6th century [2]. Notably the proof used trigonometry. Earlier similar result in West is Heron’s formula for the area of triangle. However the proof doesn’t use any algebra.

Bhaskara had devised ideas of differential calculus by 11th century [2]. A concrete example of this is the use of sine and cosine series in Kerala School of Mathematics [3].

This is really just the tip of the iceberg because majority of Indian mathematics is simply lost. What remains is poorly understood. But when reliable translations are available you can see that idea of using geometry and algebra interchangeably was common.

Maybe I’m biased. Curious to know what others make of this

[1] https://en.wikipedia.org/wiki/Brahmagupta%27s_formula [2] https://en.wikipedia.org/wiki/Bh%C4%81skara_II [3] https://en.wikipedia.org/wiki/Kerala_school_of_astronomy_and...

> Brahmagupta was the first to have a formula for area of cyclic quadrilateral in 6th century

That's not analytic geometry.

> Bhaskara had devised ideas of differential calculus by 11th century [2]. A concrete example of this is the use of sine and cosine series in Kerala School of Mathematics [3].

I don't think these count as analytic geometry.

Analytic geometry has a specific meaning: https://en.wikipedia.org/wiki/Analytic_geometry

Yeah read the distance and angle section. Describing geometric objects with equations in coordinate system is analytic geometry.

Anyway I’m not going to seriously discuss with you because it’s clear from your post history that your replies are avalanches of pedantry. If you think it has a specific meaning, elaborate it and identify the point of divergence or express what your view is. I don’t have an obligation to convince you.

> read the distance and angle section.

Distances and angles have been used by the Greeks since ancient times. By themselves they don't constitute analytic geometry.

> Describing geometric objects with equations in coordinate system is analytic geometry.

Yes, and the example you gave does not satisfy that definition.

Have you studied math? I know Greeks used angles and distances lol. The point is how. Did they describe them algebraically or did they do the Euclidean thing where they “take a compass and bisect and you’ll see each part is equal to this other side”. Euclid/Greeks used pure geometry. They might have had all geometric concepts. But they didn’t manipulate algebraic equations to obtain geometric insights. And yes that is analytic geometry

> 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.

From the wiki of synthetic geometry [1]:

> 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

Despite your attempt to obfuscate the issue with your wall of text, the simple fact of the matter is that you didn't give a single example of analytic geometry (i.e. coordinate geometry), which is based on the explicit use of a coordinate system.

> 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.

You use coordinate and analytic interchangeably but coordinate system is not a requirement nor equivalent to analytic geometry. It doesn’t even relate to mathematical analysis. Analysis, to the extent that it’s series and limits, also existed in India before Descartes.

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.

> coordinate system is not a requirement... to analytic geometry

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...

[3] https://www.thefreedictionary.com/analytic+geometry

Reply on news.ycombinator.com