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Topic: Units of area
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Posts: 447
Registered: 12/3/04
Units of area
Posted: Nov 7, 1995 10:46 AM
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David Fowler is correct: I have no evidence that Archimedes knew
about Babylonian notation. In fact, he probably didn't. I was
probably thinking of his "exponential" notation in the "Sand
Reckoner." Sorry for the slip.

But David Fowler's remarks about area simply reinforce my question
about units. Even if the Greeks used Egyptian fractions in their
area calculations, one must still ask: "What were the units of
area they were denoting?" It's exactly because I am aware of the
various "formulas" for area (some correct, others mostly incorrect)
known to the Egyptians and Greeks that I ask the question. If you
think the area of a quadrilateral is the product of the averages
of the opposite sides, you still can get non-integer values for
the area; however, even if everything comes out even, WHAT ARE
THE UNITS? Is there a "square cubit" and is there, in the
record, any mention of an area being, say 3 + 1/4 square cubits
or somesuch?

One of the reasons I am interested in this is because if area was
commonly measured in square units of some sort, the existence of
irrationals would be a great shock. A rectangle with base 1 and
height the diagonal of the unit square could not be decomposed
into congruent squares, hence its area could not be expressed
in terms of square units or squares 1/N th of a unit on a side
(for any N).

(Of course, this is more of a shock for us, since we use square
units all the time; in fact, that's how we think of area or
measure once we've chosen a unit. It forces us to think in terms
of limits even for the area of rectangles...)

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