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The Wikipedia article on squaring the circle has a relatively good history of their efforts on this particular problem, but it fails to mention why the Greeks were interested in this methodology on a wider basis. An editor added a bit of useful text about this, but I believe it needs more.

As I recall (dim memory here), mathematicians of the era were highly suspect of numerics, considering arithmetic to be something closer to bookkeeping than the purity of geometry which existed at another level of reality. As such, purely geometric solutions to problems were considered important. Is this remotely correct?

Can anyone offer a source that covers their thinking in this regard that might be a useful reference?

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    $\begingroup$ In Euclid, there is no concept of "area" as a number (indeed, there was no concept of what we now call "real number"). So, instead of that, given a geometric figure you would (in modern language) construct a square whose area is the same are the area of that figure. We use the Latin term "quadrature" for this. Beyond quadrature of polygons, classical results were quadrature of the parabola, quadrature of the lune of Hippocrates. $\endgroup$ Jun 12 at 13:03

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Basically, Euclidean geometry was considered the purest math since it carried fewer flaws than the proposed number theory at the time since numbers used "natural numbers" (i.e., integers), which produced glaring flaws. There was no such thing as decimal numbers, but to produce a decimal number, it would just be recognized that quantities could be described as an integer number of parts of another. For example, a "third" of an apple is one-part-of-three of a whole apple, and by multiplying this quantity by three, returns the whole apple.

However, to the horror of early mathematicians, integer numbers as a basis for math produced contradictions. The story of Hippasus comes to mind in which he proved that there are some numbers that can't be expressed as a ratio of two integers, hence the number was "irrational. Legend has it that the Pythagoreans drowned him for exposing this flaw.

https://en.wikipedia.org/wiki/Hippasus

So how do we fix this? We revert to something more fundamental than numbers: magnitudes (e.g., lines, areas, angles, apples, oranges, etc.), whose size is regardless of quantity. What I mean by this is that a magnitude can exist alone and still have size; and when alone, it's absurd to assign a value to it other than that the magnitude exists, and likewise, it's absurd to assign a quantity of it. In contrast, if we were to claim that the magnitude has size "five," then we're admitting that there is five of some other magnitude that defines the first magnitude. Furthermore, for a standalone magnitude, we can't say there are "one" quantity of magnitudes, but rather we only claim that it exists. For this reason, Greek math is said to start counting starting with "two" instead of "one," which is strange to us modern counters, but after all, what's the point of counting if there's only a single magnitude? In other words, counting and numbers only make sense once there's a plurality of magnitudes.

Once we add a second magnitude to the mix, then we can compare (i.e., =, <, >) them to each other, which then provides a structure that allows addition, subtraction, multiplication, and division. Euclid's Elements Book 1 is dedicated to establishing this foundation.

Furthermore, ratios of these magnitudes may be taken, which are valid regardless of number, but a ratio is only understood once we introduce a third magnitude. For example, the ratio A:B can exist, but it's useless by itself; however, introduce the ratio A:C and let A:B = A:C, then we can infer that B = C. Again, all this is done without number! The theory of ratio and proportion is all laid out in Elements Book 5.

It's not until Elements Book 7 that number theory is developed and is not investigated with ratios until Book 10. In the latter, concepts of "commensurable" and "incommensurable" are introduced, which are loosely interpreted as rational and irrational quantities. The idea is that, if we take two arbitrary magnitudes, the question is whether we can multiply each of them respectively in a way that they can create two new equal in size magnitudes. If the equality can be made with integer numbers of each, then the magnitudes are commensurable. If not, then they are incommensurable. For example, for magnitudes A and B, they are "commensurable" if 3A = 5B, which we can see that a ratio of two numbers can be produced: A:B = 5/3 (hence 5/3 is a rational number). However, for magnitudes B and C, it's possible that any number multiplication of each will fail to produce equality. In other words, no value for x or y integers satisfies the equality xB = yC, thus B and C are incommensurable and cannot be arranged to produce a ratio of numbers (i.e., the ratio of A and B is irrational). However, the ratio of B:C still exists and remains useful so long as we avoid equating them through any integer numbers. Therefore, it is much more logically sound to demonstrate relationships through geometry where irrational numbers can be avoided and ratios are preserved.

For suggested reading, I recommend portions of Euclid, particularly books 1, 5, and 6. I also found this article by math historian Grattan-Guinness to be helpful in providing historical context, where he discusses how our modern understanding of geometry was reframed through "algebraic-geometry" which resulted in subtle losses of information. For example, algebraic-geometry will say area is length times width, but euclidean geometry never defines area. Instead, the ratio of the areas of two rectangles are the same as the complex ratio of their lengths and widths, which takes on a broader meaning without the use of numbers. https://www.tau.ac.il/~corry/teaching/toldot/download/IGG.pdf

For these reasons above, Euclidean geometry was heavily relied upon up until algebra and the continuous number line was polished and widely accepted (it became somewhat solidified with Descartes). Since the magnitudes and their ratios were so powerful over numbers, it comes as no surprise that squaring the circle was a problem attempted to be solved by many; however, it wasn't until 1882 that Ferdinand von Lindemann proved that squaring the circle is impossible with Euclidean methods, primarily because, not only is pi irrational, it's also a transcendental number!

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  • $\begingroup$ Am I correct in thinking the argument would be something along the lines of "we know how to reduce the rectangle to a square, and this triangle to a square, so we can now compare the magnitudes of those squares - and so we'd really love to do the same with the circle"? $\endgroup$ Jun 17 at 19:11
  • $\begingroup$ If you're looking for motivation for the problem, I would say that, since both circles and squares enclose areas, and since these areas are magnitudes (i.e., can be added and subtracted), it would seem natural to find a square (with corresponding sides) that is equal in area as a circle (with a corresponding diameter or radius). Finding equal areas is common practice, however, "area" isn't directly used but is rather inferred (see I.34, VI.1, and VI.23). When these areas equal, the follow up question would be: how do sides of the square relate to the radius/diameter of the circle? $\endgroup$
    – Andrew R.
    Jun 18 at 4:45
  • $\begingroup$ You might also find the "guide" section of these two propositions interesting to your question -- they discuss the squaring the circle question: aleph0.clarku.edu/~djoyce/elements/bookI/propI45.html and aleph0.clarku.edu/~djoyce/elements/bookII/… $\endgroup$
    – Andrew R.
    Jun 21 at 22:40
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My article "Operationalism: An Interpretation of the Philosophy of Ancient Greek Geometry" https://link.springer.com/article/10.1007/s10699-021-09791-4 discusses possible motivations for the Greek preoccupation with constructions that is evident in Euclid and in the tradition of the classical problems of circle quadrature, angle trisection, and cube duplication. I argue that this stemmed from constructivist foundational concerns. Introducing objects into mathematical discourse in a non-constructive fashion runs the risk of introducing inconsistencies and hidden assumptions. For example, "let ABC be a triangle with three equal sides" is fine but "let ABC be a triangle with three right angles" is not. This shows that one cannot merely introduce objects by "letting" it be a figure with a certain property. A way to safeguard that one it only talking about non-contradictory objects (and hence to safeguard the consistency of mathematics) is to only admit objects that are constructed by rigorous and verifiable construction principles. Hence, in particular, one cannot merely "let" a square be equal in are to a circle; rather one must construct such a square.

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    $\begingroup$ This is an absolutely fascinating article. $\endgroup$ Jun 17 at 19:09

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