# Ancient Greek Definition of Area

I am looking for a source that would explain what the ancient Greeks thought the “area” of a polygon was, in a philosophical sense. It’s very easy to find sources about the history of their formulas for area, but I have a few points of confusion about their understanding of area qua area. I might have some misconceptions, but I would love a source that could teach me a little about the following questions.

1. My understanding is that the Pythagoreans had a notion of “magnitude” that was not coextensive with “number”. Famously, the hypotenuse of a right triangle with legs of length one has a “length of $$\sqrt{2}$$” in the sense of magnitude, but that this magnitude was not a number. Was there a similar notion of area and volume for polygons/polyhedra?

2. I heard from someone at some point that there was an ancient definition of area that was something like “the quantity that is preserved after cutting a polygon into pieces and reassembling”. This kind of reasoning was certainly present in early proofs of volume formulas, but I have heard it referred to as a definition of area. Is this true? If I recall, this was attributed to Eudoxus, but I could be misremembering.

Thank you! I look forward to learning more.

• Ancient Greeks had no concept of areas or volumes, they did not associate numbers to geometric magnitudes generally, and all their "area formulas" are modernizations. What they did prove was that some figure is to some other figure (say, a triangle to a square) as $m$ to $n$, where $m$ and $n$ were integers. This was done, for example, by dissecting figures into pieces and matching congruent pieces, complemented by "exhaustion" for curvilinear figures. These issues are discussed in Who calculated the volume (and surface area) of the sphere exactly? Commented May 3 at 23:50
• So, mathematics from a philosophical point of view according to the vision of the ancient Greeks... I have just the thing for you: A Commentary on the First Book of Euclid's "Elements" by Proclus (technically he died in the Middle Ages, but let's not be too picky). Commented May 4 at 16:03

I expand the comment of @Conifold. Euclid does not use real numbers in his geometry, so there is no notion of area as a number. Instead he studies an equivalence relation on plane figures which we can recognize as "having equal areas". Euclid himself just calls such figures "equal". This equivalence relation satisfies certain properties (which Euclid does not list explicitly, but they can be inferred from his proofs:

1. Congruent figures are "equal"

2. Unions and differences of "equal" figures are equal.

3. Halves of "equal" figures are "equal".

4. The whole is "greater" than a part of it.

5. If squares are "equal" then their sides are equal.

For detailed comments on this definition and how it is used I refer to the book

R. Hartshorne, Geometry. Euclid and beyond, Springer, 2000, Chapter 5.

• What is "half" a figure then? Commented May 4 at 14:46
• Hartshorne says that Euclid does not explicitly list those properties, but aren't they κοιναὶ ἔννοιαι (common notions)? Commented May 4 at 15:51
• @TorstenSchoeneberg Property 3 probably refers to propositions like I.37, in which the triangles are halves of equal parallelograms. Commented May 4 at 20:41
• Thank you so much for the thoughtful answer and the book recommendation!
– Joe
Commented May 4 at 21:09
• @Torsten Schoenberg: property 3 is used to prove that triangles with equal bases and heights are equal (have equal areas); this is derived from the corresponding statement about parallelograms in Proposition I.37. Commented May 5 at 12:19