Descartes’s Evil Genius
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It is that process that made it possible for elementary schoolchildren to reasonably be expected to cope with algebra and world history, topics largely limited to the old world's most brilliant academics.
Decartes was in the thick of that transition.
Resistance to zero and negative numbers had to evaporate once the number line was understood.
Source?
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...
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
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.
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.
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...
Edit: I found a draft from 2018, but the author asks for it not to be cited. It was published earlier this year, as: "Free Speech Skepticism". Susan J. Brison. Kennedy Institute of Ethics Journal, Volume 31, Number 2, June 2021
This kind of rhetorical gambit really bugs me. So if I find Gadberry unconvincing (or possibly even full of it) I'm close-minded and inattentive, am I?
This is the point in time when all the philosophy students in the audience begin to giggle, as they are familiar with the philosophical proposition of "cogito ergo sum", or "I think, therefore, I am".
But to explain the concept beforehand would be putting Descartes before the horse.