# Euclidean Geometry in Classical Thought - Realization or Representation?

First post! :]

This has been bothering me for a while now:

[Taken from John J. Roche's "The Mathematics of Measurement: A Critical History"]

When physico-mathematicians in the seventeenth century needed to represent their physical quantities mathematically they almost always turned to geometry rather than to numbers.

The author goes on to explain this further, saying how the Ancient Greeks had used geometrical figures to represent physical quantities rather than numbers (influencing the physicists later on) and how they were most appropriate for representation since they denoted "continuous physical quantities".

I don't understand this; what makes physical quantities continuous? Wouldn't you figure them to be discrete considering that they are quantified by units?

More importantly, is the author saying that physicists and mathematicians realized physical relationships geometrically? Or is he just saying that they merely chose to represent physical quantities and their relationships to others with geometrical figures? It seems extremely non-intuitive to view physical quantities as being segments/planes/solids instead of as quantifiable values with defined dimensions. Of course it's easier to view/realize proportionalities between physical quantities through geometrical/graphical means, but there must have been a numerically-based intuition even before this. In other words in finding a relationship between 2 quantities, there must have been a numerically-based line of thought that preceded a geometrical demonstration of it, simply through my reasoning that it is more intuitive.

I wanted to discuss this though, to see if anyone had any input/corrections to make about my thoughts here; I'd love to hear what you guys have to say :]

Here's a direct link to the book I'm referring to (Google Books). The quote stated above is on page 87, and it continues on from there.

Thank you!

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Of course I have never read the classical sources in the original, but in translation it sometime sounds like the classical thinkers did not conceive of "squaring" as an operation on number in its own right, but rather an expression for "finding the area of a square with side of a given length". No idea if that is just me or something real. – dmckee Jun 16 '12 at 17:38
I was thinking more along the lines of proportional thinking between quantities but this is interesting as well. If you read a little further in the link, the author explains that with geometrical figures as representational tools, it was then possible to compare completely "heterogeneous" quantities. This suggests that the Ancient Greeks perceived number as being a scheme for representing just quantity, completely separate from the concept of quantity of something. In other words, since physical quantities had "kinds"/"types" they couldn't be regarded as just an amount. – ProSteve037 Jun 19 '12 at 22:16