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Throughout our known history of geometry were the units representing areas and volumes always in terms of squares and cubes respectively? Take ancient Egyptian formulas as an example, the fact that their formulas are very close to ours must mean that the units representing areas and volumes were also unit squares and unit cubes, correct? I mean we can safely stamp their formulas with for example $cm$ and $cm^2$ and get the same result. And does that mean we can safely assume that they didn't measure e.g. volume by counting the number of times you can fill up an object by throwing identical fistfuls of sand into it?

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    $\begingroup$ In e.g. Euclidean geometry there were no "units of measure" $\endgroup$ Oct 27, 2021 at 14:20
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    $\begingroup$ See e.g. Early modern English area $\endgroup$ Oct 27, 2021 at 14:23
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    $\begingroup$ Exactly; there were no cm in Ancient Egypt etc. And thus there were no "length sqaured" but area. $\endgroup$ Oct 27, 2021 at 14:58
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    $\begingroup$ No. It was often more useful to tie volumes to weights (liter) or standard containers (amphora), and area to agricultural labor (carucate) than to length, see List of obsolete units of measurement. We are still using liters and gallons. And measuring volume by using sand or water to fill it up is actually quite effective. $\endgroup$
    – Conifold
    Oct 27, 2021 at 19:53
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    $\begingroup$ @Conifold - Indeed, one could argue that the definition of a liter is the volume of a kilogram of water (which works out and is not a random occurrence). $\endgroup$
    – Jon Custer
    Oct 27, 2021 at 20:42

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