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What are sources placing a length (one dimensional) in proportion to an area (two dimensional)?

The Greek geometers compared quantities of the same dimension: e.g. the area of a circle is in proportion to the area of a square with the radius as the side.

Descartes was to my knowledge the first to compare quantities of different dimensions: Segment $AB$ is in proportion to $XY$ as $PQ \cdot RS$ is to $TU$.

What examples of there of people saying, e.g.

Consider a line $OB$ and a curve. For any point $C$ on the curve, drop the perpendicular from $C$ to $BO$, and call its foot $C'$. The curve has the property that at any point $C$, the length $CC'$ [one dimensional] is in direct proportion to the area of a square with side $OC'$ [two dimensional] if and only if the curve is a parabola whose vertex is the point of tangency to $O$.

or any other type of proportion or equation between different dimensions (length, area, volume)?

Obviously this is common place in calculus. I'm looking for examples of where this idea developed, as it seems totally absent from classic geometry.

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  • $\begingroup$ There is the isoperimetric inequality (among plane figures with given perimeter, the one with largest area is the circle.) en.wikipedia.org/wiki/Isoperimetric_inequality $\endgroup$ Dec 27, 2023 at 18:00
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    $\begingroup$ @GeraldEdgar To me, it seems the isoperimetric inequality holds perimeter fixed and maximizes area. It never compares perimeter to area or sets up any proportion between the two (e.g. if we double perimeter, what happens to area?) $\endgroup$ Dec 27, 2023 at 19:17

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Al Khwarizmi, one of the first written sources of the general solution to quadratics, does exactly that:

I observed that the numbers which are required in calculating by Completion and Reduction are of three kinds, namely, roots, squares, and simple numbers relative to neither root nor square. A root is any quantity which is to be multiplied by itself, consisting of units, or numbers ascending, or fractions descending. A square is the whole amount of the root multiplied by itself. A simple number is any number which may be pronounced without reference to root or square.

In doing so, if I understand correctly, he seems to "upshift" length into area, by equating a length $x$ to an area $x \times 1$:

http://www.ams.org/publicoutreach/feature-column/fc-2020-11#three

If my reading is correct, then Al Khwarizmi, besides solving the quadratic, helped lay a foundation for analytic geometry and calculus, which depend on being able to compare e.g. the rate of change of $x^2$ compared to rate of change of $x$. This was unthinkable to the Greeks, but Al Khwarizmi seems to have opened the door.


A similar comment is made here:

Thus, Islamic mathematics allowed, and indeed encouraged at variance with the Greek tradition... the simultaneous manipulations of magnitudes of different dimensions as part of the solution of an individual problem. Thus... the solution of a quadratic equation is a “number”, rather than a “line segment” or an “area”... this approach was fundamental in developing a more abstract and general conception of number, which eventually became essential for the creation of a full-fledged abstract idea of an equation.

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    $\begingroup$ Ah! This is amazing! I'd never heard this! (or wouldn't have appreciated/understood its significance years ago!) Thank you! $\endgroup$ Dec 27, 2023 at 20:46
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    $\begingroup$ I'm not convinced of your interpretation, because in the few quotes you show here, Al Khwarizmi calls the quantities "numbers" and not lengths or areas. If you square a number you get a number again. There is no shift of physical dimension involved. I assume that was also how Al Khwarizmi used numbers. He says they square may be represented by a quadrate, but that's not the same as saying the square "is" an area. Maybe you can find other quotes that support your interpretation? $\endgroup$ Dec 28, 2023 at 13:44
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    $\begingroup$ And you cannot simply interpret a length x as an area by viewing it as x times 1. Since 1 is just a number (not a length), x times 1 is still a length, not an area. You would have to arbitrarily fix a unit length first, like the meter or inch etc. Then you could "uplift" x to an area as x times meter. But that construction depends on the arbitrary choice of unit length. $\endgroup$ Dec 28, 2023 at 13:49
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    $\begingroup$ This is an interpretation that I haven't come across before. And I think you are definitely onto something. $\endgroup$ Jan 12 at 14:50

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