My impression is that the name analytic geometry, which I understand roughly to be geometry in Euclidean space using coordinates, is not used that much anymore. We would probably classify the subject as an elementary version of real algebraic geometry these days, even though it's often absorbed into a course on multi-variable calculus. My question is, who coined the term 'analytic geometry'? And what was the sense in which they were using the word 'analytic'? If you know, it would be useful to have some detail on the meaning of the word in this mathematical context rather than philosophical generalities on the 'analytic-synthetic distinction.'
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asked Dec 16, 2022 at 10:43
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6$\begingroup$ Jeff Miller has an excellent webpage dedicated precisely to this sort of questions. $\endgroup$– WojowuCommented Dec 16, 2022 at 10:52
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1$\begingroup$ ... with a 1709 book as the first appearance. $\endgroup$– Carlo BeenakkerCommented Dec 16, 2022 at 10:57
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1$\begingroup$ Although it's already been answered in the comments, I'll mention this question seems better for HSM. $\endgroup$– LSpiceCommented Dec 16, 2022 at 11:50
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2$\begingroup$ Analytic geometry is not a part of algebraic geometry since it also treats the sets (in $R^n$ or in $C^n$) determined by transcendental equations. $\endgroup$– Alexandre EremenkoCommented Dec 16, 2022 at 15:04
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1$\begingroup$ @AlexandreEremenko Fair enough. I wonder if it was like that in the 17th century? It seems to me that in most conceptions, it deals with definable sets in some fairly restrictive structures. It's from that perspective that the geometry doesn't seem very 'analytic' from a present-day perspective. $\endgroup$– Minhyong KimCommented Dec 16, 2022 at 22:52
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1 Answer
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Analysis means breaking apart; taking something complex and decomposing it into simpler constituents.
This is associated with "working backwards": starting with a complicated result and finding simpler ones from which it follows. The Greeks used "analysis" in this sense in mathematics. In this process one assumes a sought result as if it was given, and works "backwards" to uncover from which simpler things it can be derived, with the intention of then reversing the steps to give a direct synthetic (synthesis = putting together) proof of the sought result.
In the 17th century, "analysis" came to mean "working with x" so to speak, because when we call a sought quantity x and start manipulating it in equations then we are indeed treating the sought as if it was known, which is exactly the classical meaning of analysis.
With the advent of calculus, since "analysis" meant "working with x" it also became associated with "working with f(x)", and hence we get analysis in today's sense of real analysis.
