The use of axes that are not perpendicular was common in very old (before 1900) analytic geometry textbooks and conics textbooks, where it's often called (in texts written in English) "oblique coordinates". To be more correct, "oblique coordinates" doesn't mean we exclude perpendicular axes, only that the axes don't have to be perpendicular (the same way that "rectangle" doesn't exclude "square"). I learned about this because of something I was working on a few years ago, in which I wound up looking at over 100 pre-1900 textbooks on analytic geometry and/or conics, mostly digitized online, but I also looked at several dozen old books available at a large university library near me. You can locate many such books that use oblique coordinates by following the google searches I posted at

The Pythagorean Theorem comes to mind. The distance between points can be found without resorting to an ugly law of cosines.

ReplyDeleteThe use of axes that are not perpendicular was common in very old (before 1900) analytic geometry textbooks and conics textbooks, where it's often called (in texts written in English) "oblique coordinates". To be more correct, "oblique coordinates" doesn't mean we exclude perpendicular axes, only that the axes don't have to be perpendicular (the same way that "rectangle" doesn't exclude "square"). I learned about this because of something I was working on a few years ago, in which I wound up looking at over 100 pre-1900 textbooks on analytic geometry and/or conics, mostly digitized online, but I also looked at several dozen old books available at a large university library near me. You can locate many such books that use oblique coordinates by following the google searches I posted at

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